Abstrac&--This paper develops expressions for the stress intensity factor and energy release rate at the existing notch tip of a three-point bend specimen. These expressions are based on the true state of stress at the notch tip and are therefore likely to give an improved estimate of the fracture toughness. Their form is as simple as that of the ASTM expressions. The fracture toughness, computed on the basis of the complex state of stress existing at the notch tip, is shown to be higher than that computed on the assumption of a simple, tension field.
THE ASTM formulae[l-31 for the determination of fracture toughness from three-point bend specimens have been extensively used for testing metallic and non-metallic materials. It is implied in these formulae that the state o$ stress immediately ahead of the pre-crack front is one of pure tension. In consequence, the crack front ought not to deviate from its initial (straight) path. Any small deviation from the straight path is unlikely to make a substantial difference to the resulting fracture toughness and may therefore be ignored. There are many instances, however, where the deviation may not be insignificant. This is likely to be so in the case of materials such as concrete and reinforced ceramics[4, 51, not only because of their heterogeneous nature but also because of the presence of a complex state of stress immediately ahead of the pre-crack front.
The true stress state at the tip of a pre-crack in a three-point bend specimen consists of not only a tensile stress normal to the crack faces but also of a significant (tensile) stress in the plane of the crack and of a shear stress. An appreciation of the relative magnitudes of these stresses may be gained from typical results of elastic finite element calculations reported in Section 2 below. Now, it is well known[G-IO] that the in-plane normal stress component controls the stability of the crack growth (a tensile in-plane stress enhances this growth, whereas a compressive in-plane stress retards it), whilst the shear stress component forces the crack front to deviate from its straight path, producing a kinked/curved crack front. It is, of course, true that in heterogeneous materials further kinking/curving of a crack front may result from the advancing crack front having to by-pass hard, second-phase particles lying along its path.
It seems appropriate therefore to modify the ASTM formulae for three-point bend specimens by making an allowance for the true stress state ahead of the pre-crack front. This has been achieved in the present work. Similar modification may be possible for compact-tension specimens, but this is not pursued here.
The modification has been accomplished by calculating the stress intensity factor and energy release rate not at the tip of the straight crack itself but at the tip of an infinitesimally small kink that is likely to form at the crack front in the presence of the complex state of stress. The resulting compliance functions are shown to depend explicitly on both the notch/depth and the span/depth ratios. It is moreover shown that the energy release rate is no longer simply proportional to the square of the stress intensity factor. Nonetheless, the compliance functions proposed here retain the simplicity of the functions in the original ASTM formulae, and it is hoped that they will find widespread acceptance.
2. PRELIMINARY CONSIDERATIONS
An appreciation of the true state of stress g,, ovy and CF_, may be gained from Table 1 which shows the typical results of elastic, finite element calculations for two three-point bend specimens. With the origin at the tip of the existing crack (depth, a), the x-axis coincides with the direction of applied load (P = 10 N) and the y-axis is directed along the span of the beam.